Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite.

Opening

For a detailed description of the Number Talk procedure, please refer to the Number Talk Explanation. For this Number Talk, I am encouraging students to represent their thinking using an array model.

Task 1: 56/8

For the first task, some students decomposed the 56 into 40+16 while others decomposed the 56 into 24 + 16 +16 or into 28 + 28: 56:8.

Task 2:112/8

During the next task, students realized that 112 is just 56 doubled. Many students doubled the 7 x 8 array to get 14 x 8: 112:8. Other students experimented with decomposing 112 into multiples of of 7: 14 x 8 : 7 = 16.

Task 3:800/8

For this task, most students made an 8 x 100 array that equaled 800.

Task 4:912/8

For the final task, students added together the quotients for task 2 and task 3: 912:8.

Throughout every number talk, I continually model student thinking on the board to inspire other students. This also requires students to use math words to explain their thinking instead of relying on a model to represent the math. As students solved each task, I wrote the answers on the board to encourage students to use prior tasks to solve the more complex tasks: Listed Tasks.

Number Talk Explanation.pdf

Listed Tasks.jpg

56:8.jpg

112:8.jpg

14 x 8 : 7 = 16.jpg

912:8.jpg

Teacher Demonstration

30 minutes

Goal & Introduction

I began today by reminding students of our learning goal: I can determine if a number is a multiple of given number. I explained: Over the past week, you have been investigating multiples of numbers up to 10. Like great mathematicians, you have come up with and tested your own conjectures about multiples as well. For example, many of you drew the conclusion that all multiples of two are even. Others of you found that all multiples of 6 are also multiples of 3. Today, I'd like for you to continue thinking about these conjectures and if they really are true!

Modeling the Game

To ensure that students would be able to successfully play the game, Multiple Turn Over (found at this link), I took plenty of time to slowly model and explain each step. I invited students to sit on the carpet at the front of the room. Some students chose to stay at their desks during this time as well. I projected the Recording Sheet on the front white board and explained the following Rules. I also asked for a couple student volunteers to model the game: Modeling the Game.

1. The first step is to deal out ten Multiple Cards to each player. For the first game, you'll only play with the Game 1 Cards. I pointed to the board where there was a stack of Game 1 Cards, Game 2 Cards, Game 3 Cards, and Game 4 Cards. I explained: When you get to Game 2, you'll mix Game 1 and Game 2 cards together. When you get to Game 3, you'll mix Game 1, Game 2, and Game 3 cards together. Finally, when you get to Game 4, you'll mix all the cards together from all the games!

I dealt each student who volunteered to model the game 10 cards. They recorded these multiples on their recording sheets.

2. Next, players arrange their Multiple Cards face up in front of them. Each player should be able to see everyone's Multiple Cards.

3. The player with the smallest multiple begins. This player calls out any whole number (except 1). Each player records that factor on his or her Recording Sheet.

4. All the players (including the player that called out the number) search for cards in their set that are multiples of that number. They write those multiples on their recording sheet and turn those cards facedown. If a player has no multiples of a number called, that player writes "none" under "Multiple Cards I Turned Over."

To model this, I asked one of the volunteers to call out a number. The student said, "4!" Altogether, as a class, we looked at each multiple card (for each player) to determine if the number was a multiple of 4: Crossing off Multiples for 4. In this video, you'll hear students using some strategies that don't always work. Looking back, I should have asked them: Does that always work?!

5. Players take turns calling out numbers. This game is over when one player turns over all ten Multiple Cards.

I also explained that students might need to use prime factorization in order to identify whether or not a number is truly a multiple of a given a factor. Here, a student proved that 54 is a multiple of 3: Factor Tree for 54.

Modeling the Game.jpg

Rules.pdf

Recording Sheet.pdf

Game 1 Cards.pdf

Game 2 Cards.pdf

Game 3 Cards.pdf

Game 4 Cards.pdf

Crossing off Multiples for 4.mov

Factor Tree for 54.mov

Student Practice

Assigning partners is always easy in my classroom as I already have students strategically placed in desk groups based upon ability levels, communication skills, and behavior. I asked each group to get a set of Game 1 Cards as well as a Recording Sheet for each partner.

Getting Started

To begin, I asked students: What is the most efficient way to cut these cards? One student offered, "Cut the card sheet in half so that both partners can cut." Another student said, "Cut the sheet into strips. Then cut card off at a time." Students immediately began Cutting Cards. Next, they shuffled the cards and dealt 10 cards to each student.

Playing the Game

Next, students began Recording their Multiples at the top of their recording sheets. Once students began playing, I conferenced with each pair of students multiple times to ask questions in order to monitor understanding and to challenge students to explain their thinking. By asking students to explain their thinking, I knew they would also be engaging in Math Practice 3: Construct Viable Arguments.

What have you noticed about all of your multiple cards?

What factor did you choose and why?

What might you do differently during the next game?

Have you noticed any challenges?

Conferences

I loved watching students play this game! More than anything, I really enjoyed students using precise vocabulary to explain their thinking (Math Practice 6: Attend to Precision): Prime Numbers.